an age-structured extension to the vectorial capacity model

An Age-Structured Extension to the Vectorial CapacityModelVasiliy N. Novoseltsev1, Anatoli I. Michalski1*, Janna A. Novoseltseva1, Anatoliy I. Yashin2,

James R. Carey3, Alicia M. Ellis4

1 Institute of Control Sciences, Moscow, Russia, 2Duke University, Durham, North Carolina, United States of America, 3Department of Entomology, University of California

at Davis, Davis, California, United States of America, 4 Fogarty International Center, National Institutes of Health, Bethesda, Maryland, United States of America

Abstract

Background: Vectorial capacity and the basic reproductive number (R0) have been instrumental in structuring thinkingabout vector-borne pathogen transmission and how best to prevent the diseases they cause. One of the more importantsimplifying assumptions of these models is age-independent vector mortality. A growing body of evidence indicates thatinsect vectors exhibit age-dependent mortality, which can have strong and varied affects on pathogen transmissiondynamics and strategies for disease prevention.

Methodology/Principal Findings: Based on survival analysis we derived new equations for vectorial capacity and R0 that arevalid for any pattern of age-dependent (or age–independent) vector mortality and explore the behavior of the modelsacross various mortality patterns. The framework we present (1) lays the groundwork for an extension and refinement of thevectorial capacity paradigm by introducing an age-structured extension to the model, (2) encourages further research onthe actuarial dynamics of vectors in particular and the relationship of vector mortality to pathogen transmission in general,and (3) provides a detailed quantitative basis for understanding the relative impact of reductions in vector longevitycompared to other vector-borne disease prevention strategies.

Conclusions/Significance: Accounting for age-dependent vector mortality in estimates of vectorial capacity and R0 wasmost important when (1) vector densities are relatively low and the pattern of mortality can determine whether pathogentransmission will persist; i.e., determines whether R0 is above or below 1, (2) vector population growth rate is relatively lowand there are complex interactions between birth and death that differ fundamentally from birth-death relationships withage-independent mortality, and (3) the vector exhibits complex patterns of age-dependent mortality and R0,1. A limitingfactor in the construction and evaluation of new age-dependent mortality models is the paucity of data characterizingvector mortality patterns, particularly for free ranging vectors in the field.

Citation: Novoseltsev VN, Michalski AI, Novoseltseva JA, Yashin AI, Carey JR, et al. (2012) An Age-Structured Extension to the Vectorial Capacity Model. PLoSONE 7(6): e39479. doi:10.1371/journal.pone.0039479

Editor: Andrew J. Yates, Albert Einstein College of Medicine, United States of America

Received March 9, 2012; Accepted May 23, 2012; Published June 19, 2012

Copyright: ! 2012 Novoseltsev et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The research was supported by the California Department of Agriculture and Natural Resources, National Institute of Aging/National Institute of Healthgrants P01 AG022500-01 and P01 AG08761-10. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of themanuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: [email protected]

Introduction

The basic reproductive number (R0) and vectorial capacity areintegral parts of the language, science, and control of vector-bornedisease [1]. For almost 100 years, since the initial work by Ross [2]through the contributions by Macdonald [3] and Garrett-Jones [4]into the present, deviation from simplifying assumptions of themodels has rarely been examined. Important assumptions are thatvectors do not senescence, vector populations are homogeneous,size of adult vector populations is constant, vector bites aredelivered at random and uniformly, and infected vectors never losetheir infectiousness [1]. Nowadays many basic assumptions arebeing revised. Because of climate change the influence oftemperature and its variations on malaria transmission isinvestigated thoroughly [5,6,7]. The departure from assumptionthat vectors do not senescence is discussed in [8,9]. All theseconsiderations are important for development the intervention

strategies to control vector transmitted diseases [10,11]. In order tobetter understand how variation in one of the most sensitivecomponents of a vector’s role in pathogen transmission effecttransmission dynamics and estimation of R0, we explored theconsequences of a shift from age-independent to more biologicallyrealistic age-dependent vector mortality. In distinction to previousworks [8,9] we consider a set of models for vector mortality, whichreflect different change of mortality with age. They are Gompertzmodel with exponential growth of mortality, which is traditional indemography, logistic model with mortality leveling at advancedage as in heterogeneous population, models with declining, U-shaped and hump-shaped dependencies of mortality on age. In thepaper we derive the general form for vectorial capacity in agingvector population. Parameterization of the survival function S x! "in such population gives expression for vectorial capacity for anyvector mortality model and can be used for any vector controlscenario.

PLoS ONE | www.plosone.org 1 June 2012 | Volume 7 | Issue 6 | e39479

The basic reproductive number describes the potential forspread of an infectious disease. It defines the average number ofsecondary cases that arise from the introduction of a singleinfectious individual into a completely susceptible population [12].If R0.1 the pathogen will spread to other individuals and anepidemic can occur [2,3]. If R0,1 transmission will not persist.For mosquito-borne disease, R0 is given by the classic formula[12]:

R0~bc

rC

where b is the proportion of bites by infectious vectors that lead tohost infection, c is the proportion of bites on infected hosts thatlead to vector infection, 1/r is the length of time it takes tonaturally clear a host from infection, and C is vectorial capacity,defined as the number of infective mosquito bites that arisefollowing the introduction of a single infectious host into asusceptible population. Vectorial capacity describes the potentialfor a vector population to transmit a parasite [4]. It can be used toexplore the relative impact of different disease preventionstrategies, or put differently, ways to reduce R0 below 1. Forexample, an effective vaccination will decrease R0 by reducinginfection rates of hosts (b) and mosquito vectors (c).For two of the most important vector-borne diseases of humans,

malaria and dengue [13,14], vaccines are not yet available.Although anti-parasitic drugs are a component of malariaprevention, an analogous strategy is not planned for dengue; i.e.,use of antivirals for population based dengue prophylaxis [15]. Incases like this, vector control is a prominent component of diseaseprevention programs. The goal is to reduce R0 below 1 byreducing vectorial capacity (C). This can be achieved as illustratedby the classic model for vectorial capacity [12,16]:

C~ma2

ge{gn

Where m is the number of vectors per host, a is the human bitingrate, g is the mortality rate of vectors, and n is the extrinsicincubation period (Table 1). Many vector control programs useinsecticides in an effort to decrease vectorial capacity by reducingmosquito vector population densities (m) and increasing the adultmosquito mortality rate (g). Vectorial capacity can also be used toestimate threshold density necessary for sustained transmissionwhere R0=1, which provides an operational target for vectorcontrol programs. If an intervention can decrease vector densitiesbelow this level, R0 will be less than 1 and pathogen transmissionwill decrease to elimination. Although it is difficult to make preciseestimates of vectorial capacity for reasons explained by Dye [17],understanding the relative impact of different interventionstrategies on model components can provide helpful insights intothe approaches that are most likely to successfully prevent vector-borne disease in a given situation.Simplifying assumptions of the classic models for vectorial

capacity and R0 are useful approximations. Some, however, areinconsistent with mosquito biology and can constructively beviewed as starting points for more complex analyses [1].Growing evidence indicates that age-independent vector mor-tality (i.e., constant g across all adult mosquito ages) is violatedby mosquitoes in the laboratory and field [9,18–24]. Forexample, mortality rates for the principal mosquito vector ofdengue virus, Aedes aegypti, increase with age and, thus, are age-dependent [9,23–26].

The age pattern of vector mortality is epidemiologicallyimportant because it can have strong effects on the probabilitythat a vector will live long enough to become infectious andtransmit a pathogen [17]. Horizontal transmission for pathogens,like malaria and dengue, requires that a vector is exposed to thepathogen after imbibing an infected blood meal, survives anincubation period during which the pathogen multiples ordevelops so that at the end of that period the vector is infectious,and then transmits the pathogen when it bites a susceptiblevertebrate host. Incubation periods of many vector-borne patho-gens are thought to be only slightly shorter than the mean lifespanof their vector. Relatively small changes in lifespan can, therefore,result in relatively large changes in the number of vectors thatbecome capable of pathogen transmission. All other factors beingequal, transmission rates will be highest if mortality rates decreasewith vector age because the vector population will be composed ofmostly older vectors that have lived long enough to becomeinfectious. On the other hand, when vectors senesce, transmissionis expected to decrease because the population age-distributionwill shift to younger vectors, most of who will not live long enoughto become infectious. For these reasons, understanding underlyingpatterns of mortality is of fundamental importance for effectivedesign and implementation of vector control strategies. This ishighlighted in the conceptual development of strategies that targetthe older, potentially infective portion of vector populations[27,28]. In this paper we develop and investigated new models forR0 and vectorial capacity that take age-dependence of vectormortality into account.

Materials and Methods

BackgroundStyer et. al [9] modeled total population vectorial capacity (Ct)

for Ae. aegypti as a function of age-specific vectorial capacity, Cx,(i.e., daily number of potentially infective bites resulting from amosquito of age x biting one infectious host) and the age structureof the population, Vx:

Ct~Xv

x~s

Cx|Vx# $~Xv

x~s

ma2 Pxzn

i~xpi|exzn

! "|Vx

# $!1"

where s is the age at which mosquitoes begin biting hosts, v is theoldest biting age class, Vx is the fraction of the mosquitopopulation in age class x, m is the number of vectors per host, ais the number of host specific bites per vector per day, n is theduration of the extrinsic incubation period, pi is daily survival atage i, and ex+n is the expectation of remaining infectious life at agex+n (Table 1). Styer et al. [9] conducted a laboratory study todetermine the pattern of age-dependent mortality for Ae. aegypti(i.e., exponential, logistic, or Gompertz’s mortality models) andperformed simulations of equation 1 using parameter estimates forbest fit mortality models. They demonstrated that Ae. aegyptisenesce and that senescence produces different estimates ofpopulation vectorial capacity than when assuming age-indepen-dent vector mortality.Although Styer et al.’s [9] model was useful in demonstrating

how senescence can influence the dynamics of mosquito-bornediseases, it is not general enough to be easily applied to othervector species (mosquitoes other than Ae. aegypti as well as othernon-mosquito insect vectors) with different mortality patterns. Toaddress this issue we built on the Styer et al. [9] model in threeways. We derived (1) equations for the age structure of themosquito population, Vx in equation 1, so that it can be used forany vector and different patterns of age-dependent mortality, (2) a

Age-Structured Vectorial Capacity Model


new equation for R0 that incorporates age-dependent vectorialcapacity, and (3) a new equation for the proportion of the hostpopulation that is infectious when vector mortality is age-dependent. We illustrate the behavior of the model usingparameter estimates from Styer et al. [9] for Ae. aegypti anddifferent models of age-dependent mortality.

Vectorial Capacity in Vector PopulationBased on equation (1), we modeled total vectorial capacity of a

population of mosquitoes, C. The derivation of the formula isgiven in Supplement S1. Some assumptions were made in thederivation. We assume that age structure of the mosquitopopulation are constant over time and that age-dependentfecundity and mortality rates do not vary across time. Theresulting expression for the vectorial capacity of a population ofmosquitoes is given by:

C~ma2

%v

0

S t! "e{rtdt

&v

s

e{rx

&v

xzn

S t! "dtdx

where S x! " is the survival function of mosquitoes at age x, r isthe intrinsic population growth rate of the mosquitoes popula-tion, and v is the maximal life span. This formula gives thegeneral form for vectorial capacity in aging vector population. Itcaptures two important features of the population. First, it canbe used for any mortality model in vector population, whichproduces the specific form for survival function S x! ". In thecase of age-independent mortality survival function is given bythe classical expression S x! "~e{gx, where g is mortality rate.Presented formula allows calculating vectorial capacity of a

Table 1. Parameter descriptions.

Parameter Description

R0 Basic reproductive number, the number of secondary cases that arise from the introduction of one infectedhost into a completely susceptible population

B Proportion of bites by infectious vectors that lead to host infection

C Proportion of bites on infected hosts that lead to vector infection

1/r Duration of viremic period in host

C Vectorial capacity, number of infective bites that arise following the introduction of a single infectious hostinto a completely susceptible population

M Number of vectors per host

N Total number of mosquitoes

A Biting rate

G Age-independent mortality rate of vectors

N Duration of extrinsic incubation period

Cx Age-specific vectorial capacity

Vx Fraction of the vector population in age class x

S Minimum age at which vectors begin biting hosts

V Maximum vector life span

pi Daily survival at age i

ex Remaining life expectancy at age x

dx Age distribution

R Intrinsic growth rate

S(x) Survival at age x

g(x) Age-specific mortality rate of vectors at age x

e(x) Fecundity at age x

H Entropy; describes curvature of survival curve

A Initial mortality rate

B Rate of exponential mortality increase with age

Q Parameter influencing location of age-dependent mortality extrema

m Parameter of age-dependence in mortality

J Parameter of age-dependence when mortality declines with age

e0 Mean lifespan

Y(t) Proportion of the vector population that is infectious

X(t) Proportion of the host population that is infected and infectious

pt(x) Probability that vector of age x born at time t is infectious

et(y) Probability that vector born at time t becomes infected during 1st blood meal at age y

E Emergence rate of vectors

doi:10.1371/journal.pone.0039479.t001



vector population in the case of any parametric or nonpara-metric model for age-dependent mortality. In this case

S x! "~e{%x

0

g t! "dt, where g t! " is age-dependent mortality rate.

Second, the formula accounts the possibility of changing in sizeof the vector population. To avoid the additional hypothesisabout changing in the age structure of the population welimited consideration by the case of stable population [29]where the age structure completely defined by the survivalfunction. Table 2 summarizes formulas for vectorial capacitywith age-independent and age-dependent mortality for stable (rand age structure are constant over time) and stationary (r=0,age structure is constant over time) vector populations.

Role of Age-dependent Mortality in Determining R0Age-dependent mortality can have important implications for

the spread of vector-borne pathogens through host populations.We investigated this by simulating host and vector dynamics andcalculating R0 for Ae. aegypti. A similar exercise could be done forother vector species.Supplement S2 presents equations for the proportion of the host

population that is infected or infectious when vector populationexhibits age-dependent mortality. These equations were used in aspecies-specific example to simulate dynamics of dengue trans-mission in a human population by Ae. aegypti. We used the logisticmodel for age-dependent mortality:

g x! "~aebx 1zas

bebx{1' (# ${1

(parameters are defined in Table 1) and the parameter estimatespresented in Styer et al. [9] (i.e., a=0.0018, b=0.1416,s=1.0730).To calculate the basic reproductive number, R0, in a stable

vector population with age-independent mortality we used theclassic expression [12]:

R0~mbca2

rge{g nzs! "

with parameters a=0.75, b = c = 0.5, g=1/32, r=0.01, n=10,s=3 adopted from Styer et al. [9]. The general relationshipbetween the basic reproductive number and the vector capacity is

given de R0~bc

rC from which the basic reproductive number

R0in stable population with age-dependent mortality is given by:

R0~mbc a2

r%?

0

S t! "dt

&?

s

&?

xzn

S t! "dtdx

We use the same parameter values as above and the survival

function for logistic mortality S x! "~ 1zas

bebx{1' (# ${1=s

to

explore the dynamics of these models.

Results

Vectorial Capacity for Different Patterns of Age-dependent MortalityTo examine the behavior of the model, we calculated

vectorial capacity for Ae. aegypti as an example using parameterestimates from Styer et al. [9] for age-independent (i.e.,exponential), Gompertz, and logistic mortality models (Fig. 1)with different values of intrinsic population growth, r. In allcalculations maximal life span was set to infinity (v~?). Forthe age-independent mortality exponential model, vector capac-ity is negatively related to r across all values of r examined(Fig. 2). Vectorial capacity differs most between age-independentand age-dependent mortality models at intermediate and low rvalues. If age-independent mortality increases, the correspondingvectorial capacity curve goes down and can intersect the curvesfor age-dependent mortalities but at any case vectorial capacityfor age-independent mortality decreases with an increase ofintrinsic population growth parameter r. When, however, thevector exhibits age-dependent mortality characterized by thelogistic or Gompertz models, vectorial capacity exhibits aunimodal relationship with r (Fig. 2).Next we evaluated how age-dependent mortality affects

vectorial capacity of a stationary vector population (r=0) withGompertz, declining, U-shaped, and unimodal age-dependentmortality trajectories (Table 3). We did not test the logisticmortality model (Table 3) because with the parameter estimates inStyer et al. [9] it gives vectorial capacity values close to theGompertz model (Fig. 2). Although we were primarily interested inexamining the difference between the mortality models, there canbe considerable variability in the form of the survival functionwithin each mortality model. The unimodal mortality model, forexample, can have variation in the age at which the maximummortality occurs and the overall shape of that relationship.To better examine the full range of behavior within each

mortality model we present vectorial capacity as a function of lifetable entropy, H [30]. This value generalises behaviour of thesurvival curve and allows comparison of different curves withoutconsideration of the corresponding parameters. Entropy describesthe curvature of the survival curve (Fig. 3) and can be interpretedas the percentage change in life expectancy produced by a 1%decrease in the force of mortality at all ages [30]. Entropy is givenby:

H~{1

e0

&?

0

S x! " lnS x! "dx

where e0 is mean life span given by e0~%?

0

S x! "dx [31] and S(x) is

survival at age x. It is clear that for any survival curve H is not

Table 2. Formulas for vectorial capacity in stable andstationary vector populations.

Population Type of Mortality Vectorial Capacity

Stable Age-independent C~ ma2

g e{gn{ rzg! "s

Age-dependentC~ ma2%v

0

S t! "e{rtdt

%v

se{rx

%v

xzn

S t! "dtdx

Stationary Age-independent C~ ma2

g e{g nzs! "

Age-dependentC~ ma2%v

0

S t! "dt

%v

s

%v

xzn

S t! "dtdx




negative. In the case of a non-aging population, mortality with ageis constant giving a concave, exponentially decreasing survivalcurve (proportion surviving vs. age) and H=1 (Fig. 3). For

rectangular survival curves where all individuals die at exactly thesame age, H=0. Finally, when there is very high infant mortalityfollowed by relatively higher survival rates for older individuals,

Figure 1. Illustration of mortality models examined with different parameter values. Parameter values not listed; Gompertz: a= 0.01,declining: m=0.7, g=0.1, logistic: a= 0.007, s= 0.2, U-Shaped: m=0.001, e0= 24, unimodal: g=0.05, e0= 30.doi:10.1371/journal.pone.0039479.g001



H.1. Entropy is thus a continuum that represents different formsof the survival function (Fig. 3) and is directly related to the patternof age-dependent mortality.Figure 4 presents vectorial capacity as function of H for

stationary vector populations under fixed values of mean life span,e0, for different mortality models. For the Gompertz model(Fig. 4A, Table 3), age-independent mortality, and maximumvectorial capacity for a given lifespan occurs when b, theexponential mortality increase with age, equals 0 and H=1. Witha fixed value for vector lifespan, decreasing H and increasing bgenerates lower vectorial capacity (Fig. 4A). An increase in bcorresponds to an increase in mortality rate across all ages (Fig. 1)or proportionally greater increase in mortality for older vectorsthat would have lived long enough to become infectious thanyounger vectors that are not old enough to have completed thepathogens incubation period (Fig. 1). For a given value of H,vectorial capacity decreases with decreasing lifespan. A shorterlifespan reduces the probability that a vector will become infectedduring its lifetime and if infected, reduces the number of times thata vector can transmit a pathogen to susceptible hosts.

For a declining mortality rate (Fig. 4B, Table 3), age-independent mortality occurs when parameter j=1 (H=1). Incontrast to the Gompertz model, age-independent mortality givesthe lowest vectorial capacity. For a given value of mean life span,vectorial capacity decreases as H decreases and j increases. Ingeneral, the decrease in vectorial capacity with increases in j islikely due to an overall increase in the population mortality rate(Fig. 1).For a U-shaped relationship between vectorial capacity and age

(Fig. 4C, Table 3), the minimum of the mortality function islocated at xmin= qe0. Changing q changes the location of thisminimum (Fig. 1). For age-independent mortality, m= 0 and, as forthe Gompertz model, gives the highest vectorial capacity for thismodel. For a given mean lifespan, vectorial capacity decreaseswith a decrease in q and H and an increase in g (Fig. 4C) becausethis increases overall mortality (Fig. 1).With unimodal mortality (Table 3), the H-dependence of

vectorial capacity differs fundamentally from the cases describedabove (Fig. 4D). In our calculations, we varied parameters m and g.The location of the maximum mortality was determined asxmax~ve0. For age-independent mortality (H= 1), vectorial

Figure 2. Vectorial capacity in a stable population for three mortality models (see Table 3 for functions). Parameters used incalculations are: exponential (dotted line, g= 0.0313), Gompertz (dashed line, a= 0.00662, b=0.06234), and logistic (solid line, a= 0.00662,b= 0.06234, s= 1.073); taken from Styer et al. 2007a.doi:10.1371/journal.pone.0039479.g002

Table 3. Mortality models used in this study.

Mortality Model Function

Exponential (age-independent) g(x) = g

Gompertz (increases with age) g x! "~aebx

Logisticg x! "~ aebx

1z asb ebx{1! "

Decline with Age g x! "~mkxzg, kv1

U-shaped g x! "~mx2{kxzg, k~2qme0

mw0, qw0, g§k2=4m

Unimodal g x! "~max {m x{ q{1=4! "e0! " x{ qz1=4! "e0! ",g# $,mw0, qw0:25, gƒ1=e0




capacity is at a minimum. In contrast to the case with U-shapedmortality, any deviation from age-independent mortality increasesvectorial capacity. For a given lifespan, vectorial capacity can bepositively or negatively related to H depending on the value of q.As H decreases, older vectors have a higher probability ofsurviving (Fig. 2). However, changes in q change the age at whichmortality is at a maximum (Fig. 1). This is important because theage when vector mortality is at a maximum determines if mortalityis relatively high or low for age groups best able to transmit apathogen. The interaction between these 2 processes generates thecomplex pattern for a given lifespan.Across all mortality models examined, mean lifespan is an

important parameter determining vectorial capacity (Fig. 4). Fordecreasing, U-shaped, and unimodal mortality models, otherparameters are also important. This is particularly true for theunimodal mortality curve where vectorial capacity is sensitive tochanges in the parameter determining the location of the mortalitymaximum.

Role of Intrinsic Growth Rate rFigure 2 shows vectorial capacity in a stable population in

dependence on intrinsic rate of grows r for three mortality models:exponential, Gompertz and logistic. The largest differences invectorial capacity between these models occurs when value of r isintermediate or low. For age-independent mortality, lower r valuescorrespond to lower birth rates because the mortality rate isconstant. In this case, decreasing r by decreasing birth rates causesan increase in the proportion of the vector population in older ageclasses that are capable of transmitting a pathogen, which in turnincreases vectorial capacity. Conversely, with age-dependentmortality age-specific mortality and fecundity rates vary non-linearly with different values of r. The complex interactionbetween nonlinear birth and death rates and r lead to a decreasein vectorial capacity as r decreases, which can occur because meanage and thus transmission potential decrease as r decreases. Thisreinforces the need for new data that can be used to refine and testpredictions regarding associations between intrinsic rate of growthand the complexity of age-dependent mortality.

R0 and Dynamics of Host InfectionDepending on mosquito population density, there are epidemi-

ological important differences in dynamics of human andmosquito infections with age-dependent versus age-independentvector mortality (Fig. 5). When mosquito density is relatively high(m= 1.5), using the logistic mortality model and parameterestimates obtained by Styer et al.’s [9] mortality study, the basicreproductive number with age-dependent mortality is R0~167:3and R0~449:6 for the age-independent mortality. For such high

values of Ro, epidemic levels of transmission would be expected forboth age-independent and -dependent mortality and stationaryvalues would be similar for infectious host and vector populations(Fig. 5A,B). A different type of dynamics occurs when using thesame parameter estimates, but with a low mosquito density,m=0.007. With age-dependent mortality R0~0:8 and theproportion of infected vector and host populations declines withtime (Fig. 5C,D). For age-independent mortality R0~2:1 and theinfected proportion of the population increases over time.

Discussion

The new mathematical models derived and investigated in thispaper extend the theoretical and empirical understanding of age-dependent vector mortality in ways that can add to the conceptualbasis of vector-borne disease prevention. The formula fordependence of vectorial capacity from the vector survival isgeneral enough and can be used in calculation of vectorial capacityfor any of age-dependent mortality of vectors. One immediateutility of the formula is in the case, when a vector is exposed to twocauses of death: exogenous, which does not depend on the vectorage, such as predation, swatting, weather conditions, andendogenous, related to ageing and vector senescence. It is realisticto consider these two causes as independent hazards, whichmathematically means that the resulting mortality is a sum of theexogenous and endogenous mortalities. The presented formulaallows calculations of vectorial capacity to find the limits ofcontrol, when measures, aimed to increase the exogenousmortality, can be effective against the background of endogenousage-dependent mortality. In this case the value of age-independent

Figure 3. Hypothetical survivorship curves for different values of H, entropy.doi:10.1371/journal.pone.0039479.g003



mortality component can be considered as a parameter forconstruction of optimal control strategy under given resourcelimitations.Introduction of age-dependent mortality broadens and refines

the vectorial capacity paradigm by introducing an age-structuredextension to the model. It encourages further research on theactuarial dynamics of vector populations and the contribution ofdynamics in vector mortality to dynamics in pathogen transmis-sion. It provides a quantitative basis for understanding the relativeimpact of reductions in vector longevity compared to otherstrategies for prevention of vector-borne disease; i.e., various formsof vector control, vaccines, and anti-pathogen drugs that areapplied separately or in combination.Our analysis indicates that different age-dependent patterns of

mortality can influence vectorial capacity differently. Althoughmean lifespan remains an important determinant of vectorialcapacity, other parameters that affect the shape of a mortalitycurve are also important. We used demographic entropy toillustrate such influence in one value, which is free from thespecific parameterization of age-dependent mortality. In additionto mean life span, our analysis of entropy illustrates how survivalcurves can differ from the original assumption of exponential, orage-independent, mortality profile.

The shift from age-independent to age-dependent mortality canbe viewed as conceptually advantageous because it capturestransmission dynamics in a more biological relevant way [9]. Italso increases the mathematical complexity of the models, whichraises questions about their general applicability in applied,epidemiologic contexts. A key issue in this regard concerns thestrength of the insights gained from the increased complexity,biological and mathematical. That is, does including the additionalage-dependent elements substantially improve the power of themodels for guiding disease surveillance and prevention? Resultsfrom our analyses indicate that there are important differences inthe epidemiologic output from age-dependent vs -independentvector mortality models. Understanding the nature of outputdifferences is challenging because it depends on variation incomplex characteristics of the system being examined. Examplesinclude novel vector control strategies that target older vectors oraim to shorten vector lifespan [27,32] or prevent disease bypathogen elimination [33].Our vectorial capacity model has several limitations. It assumes

no emigration or immigration of vectors or hosts, which can beparticularly important if rates are not equal. It is deterministic anddoes not account for individual variatiability (hidden heterogene-ity) in chances of survival. In some cases it may be difficult to fit

Figure 4. Vectorial capacity versus entropy H for different values of mean life span e0. (A) Gompertz’s, (B) declining, (C) U-shaped, and (D)unimodal mortality models. For each model, age-independent mortality for each life span is noted by an open circle.doi:10.1371/journal.pone.0039479.g004



simple mortality functions (e.g., Gompertz, Weibull, etc. [30]) tofield or laboratory data [34]. More research is needed to examinethe impact on vectorial capacity when vector age-structure andpopulation growth rate vary over time.Estimates of vectorial capacity, and thus R0, for mosquito-borne

pathogens differ most between cases of age-independent and age-dependent mortality when: (1) vector densities are relatively low,(2) vector population growth rate is relatively low and there aredifferences in the complex interactions between birth and death;and (3) vectors exhibit complex patterns of age-dependentmortality. In many parts of the world vector control is an integralor even primary component of vector-borne disease prevention.Our analyses indicate that when vector control programs aresuccessful and mosquito densities are reduced to low levels, themortality model used to predict sustainability of that success or theeffort needed for the final push to eliminate the pathogen can leadto strikingly different conclusions. At the same low vector density(m=0.007), age-independent mortality predicts increasing, epi-demic transmission (Ro.1) and age-dependent mortality predictslocal pathogen extinction (Ro,1). An age-independent model mayoverestimate the effort needed to meet public health goals at lowvector densities.

Our conclusions support earlier results indicating that age-dependent vector mortality can influence transmission dynamicsand the success of disease prevention strategies in meaningful ways[9,27,35]. In a similar study, Bellan [8] demonstrated that age-dependent vector mortality has important effects on vectorialcapacity and vector control. By focusing only on the logisticmortality model and possible effects of two control measures(decreased survival and decreased recruitment that is equal acrossall ages, with each intervention affecting a single parameter) andsimplifying the equation for vectorial capacity, he demonstratedthat the effects of interventions may be over- or under-estimatedwhen assuming age-independent survival. Our study, however,goes further by providing a single complete formula for vectorialcapacity that can be used for any vector, mortality model, andcontrol scenario. Our formula allows researchers to calculatevectorial capacity and assess the effects of various control measuresfor their own particular system. Importantly, this equation may beeasily expanded to include more complex functions affectingvectorial capacity including density-dependent mortality (inaddition to age-dependent mortality) or factors affecting bitingrate.It is important to note that despite growing evidence showing

notable effects of age-dependent mortality on estimates of vectorial

Figure 5. Transmission dynamics when an infectious host is introduced at time=0. (A) age-dependent mortality, m= 1.5, (B) age-independent mortality m= 1.5, (C) age-dependent mortality, m= 0.007, (D) age-independent mortality m= 0.007. Solid line denotes humans anddashed line mosquitoes.doi:10.1371/journal.pone.0039479.g005



capacity and effects of control, we still know very little about theprevalence and pattern of age-dependent mortality in naturalmosquito populations [20,23,24,25]. Age-dependent mortality andthe diversity of its patterns has been examined for non-vectorinsects [20,34]. Most of those studies lacked sufficiently largesample sizes to accurately estimate mortality rates of relativelyrare, older-aged individuals [34]. We similarly know little aboutwhat factors govern disease vector mortality patterns or howmortality patterns vary through space and time. Theoretically,variation in patterns of age-dependent mortality could causefrequent and dramatic fluctuations in vectorial capacity andentomological thresholds below which epidemic pathogen trans-mission will cease. New techniques to better estimate patterns ofage-dependent vector mortality, how mortality patterns vary inspace and time, and the factors determining those patterns areneeded to better understand when and how age-dependent vectormortality has its greatest affects on transmission dynamics anddisease intervention campaigns.The field of aging research can make substantial contributions

to improved understanding and more efficient prevention ofvector-borne disease because it deals with factors and mechanismsaffecting age-specific patterns of mortality among different species[36,37]. Identifying concepts and interventions capable ofaccelerating vector aging processes, and understanding how suchmanipulations affect pathogen transmission parameters canstimulate investigation of new approaches for vector-borne diseasecontrol. Consideration of more realistic situations will requiremore sophisticated models and more comprehensive computa-

tional analyses of alternative scenarios. For example, undefinedheterogeneities in vector mortality patterns are likely to beimportant determinants in the success or failure of vector controlprograms. Due to variation within and between mortality patterns,a strategy that works well at one place and time may not work atanother. Our analysis indicates that a more sophisticatedanalytical framework, which is mathematically and computation-ally plausible, will stimulate increasingly insightful thinking aboutage-dependent vector mortality and prevention of vector-bornedisease. Although our analyses use data from a single mosquitospecies, Ae, aegypti, our models are intended to have broadapplication for a wide range of vector species and vector-bornediseases.

Supporting Information

Supplement S1 Vectorial capacity in stable and station-ary vector population.(DOC)

Supplement S2 Proportion of the host population that isinfected or infectious when the vector populationexhibits age-dependent mortality.(DOC)

Author Contributions

Analyzed the data: JAN AME. Wrote the paper: VNN AIM AIY JRC.

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an age-structured extension to the vectorial capacity model

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